The present invention relates to arrayed waveguide gratings (AWGs). In particular, though not exclusively, the invention concerns passband flattening in AWGs and an improvement for increasing passband uniformity in AWGs having flattened passbands.
AWGs, sometimes also known as xe2x80x9cphasarsxe2x80x9d or xe2x80x9cphased arraysxe2x80x9d, are now well-known components in the optical communications network industry. An AWG is a planar structure comprising an array of waveguides arranged side-by-side which together act like a diffraction grating in a spectrometer. AWGs can be used as multiplexers and as demultiplexers, and a single AWG design can commonly be used both as a multiplexer and demultiplexer. The construction and operation of such AWGs is well known in the art. See for example, xe2x80x9cPHASAR-based WDM-Devices: Principles, Design and Applicationsxe2x80x9d, M K Smit, IEEE Journal of Selected Topics in Quantum Electronics Vol.2, No.2, June 1996, and U.S. Pat. No. 5,002,350 and WO97/23969.
A typical AWG mux/demux 1 is illustrated in FIG. 1 and comprises a substrate or xe2x80x9cdiexe2x80x9d 1 having provided thereon an arrayed waveguide grating 5 consisting of an array of channel waveguides 8, only some of which are shown, which are optically coupled between two free space regions 3,4 each in the form of a slab waveguide. At least one substantially single mode input waveguide 2 is optically coupled to an input face 9 of the first slab waveguide 3 for inputting a multiplexed input signal thereto, and a plurality of substantially single mode output waveguides 10 (only some shown) are optically coupled to an output face 20 of the second slab waveguide 4 for outputting respective wavelength channel outputs therefrom to the edge 12 of the die 1. The array waveguides all have an aiming point C1 which is located on the output face 20 of the second slab waveguide 4 and which is the centre of curvature of the input face 15 of the second slab waveguide which input face lies on an arc of radius R. This arc of radius R is also sometimes referred to as the xe2x80x9cgrating linexe2x80x9d. The output face of the second slab waveguide lies on an arc, sometimes referred to as the xe2x80x9cfocal linexe2x80x9d, having a radius R/2 and a centre of curvature C2. The centres of curvature C1, C2 of the input and output faces 15,20 of the second slab 4 lie on a straight line X. The array waveguides 8 are all equally angularly spaced, there being a fixed angle xcex94xcex1 between neighboring array waveguides, with respect to the centre of curvature C1 of the input face 15 of the slab waveguide 4. There is also a fixed lateral spacing da between neighboring array waveguides 8, at their interface with the slab waveguide 4. The input and output faces of the first slab waveguide have a similar (but inverted) arrangement, as indicated in FIG. 1.
In generally known manner, there is a constant predetermined optical path length difference between the lengths of adjacent channel waveguides 8 in the array (typically the physical length of the waveguides increases incrementally by the same amount from one waveguide to the next) which determines the position of the different wavelength output channels on the output face of the second slab coupler 4. Typically, the physical length of the waveguides increases incrementally by the same amount, xcex94L, from one waveguide to the next, where
xcex94L=mxcexc/nc 
where xcexc is the central wavelength of the grating, nc is the effective refractive index of the array waveguides, and m is an integer number. In known manner, the transmission waveguides and slab waveguides are typically formed (e.g. using standard photolithographic techniques) as xe2x80x9ccoresxe2x80x9d on a silicon substrate (an oxide layer and/or cladding layer may be provided on the substrate prior to depositing the waveguide cores) and are covered in a cladding material, this being done for example by Flame Hydrolysis Deposition (FHD) or Chemical Vapour Deposition (CVD) fabrication processes.
In such an AWG, the passband (i.e. shape of the transmission spectrum T(xcex), which is a plot of dB Insertion Loss against Wavelength) for each output channel generally corresponds to the coupling of a Gaussian beam into a Gaussian waveguide, and is therefore itself Gaussian-shaped. In many situations it would be more desirable for the AWG to have a flat passband. This is generally because a Gaussian passband requires accurate control over emitted wavelengths, thus making it difficult to use in a system. Various ways of achieving a flat passband have been proposed, one way being to use xe2x80x9cnear field shapingxe2x80x9d. This involves creating a double-peaked mode field from the (single peak) input mode field. When this double-peaked field is convoluted with the single mode output waveguide, the resulting passband takes the form of a single, generally flat peak. One way of creating the necessary double-peaked field is to use an MMI (Multi-Mode Interferometer) on the end of the input waveguide (or each input waveguide, where there is more than one), adjacent the first slab coupler, as shown in FIG. 3(a). The MMI creates higher order modes from the single mode input signal and these multiple modes give rise to a double-peaked field at the output of the MMI.
U.S. Pat. No. 5,629,992 (Amersfoort) describes this passband flattening technique in detail. An alternative technique is to use a parabolic-shaped taper or xe2x80x9chornxe2x80x9d on the end of the input waveguide, as shown in FIG. 3(b). This is described in JP 9297228A. The parabolic taper gives rise to continuous mode expansion (by excitation of higher order modes) of the input signal along the length of the taper, until both the fundamental and second order modes are present, thus forming a double-peaked field at the output end of the taper. Other non-adiabatic multimode waveguide taper shapes can alternatively be used to achieve the desired passband flattening effect, for example a curvilinear taper shape based on a cosine curve, as described in our pending UK Patent application No. 0114608.3 the entire contents of which are incorporated herein by reference.
Near-field shaping to produce the desired multiple peak field at the input to the first slab waveguide can also be achieved using other techniques such as a Y-branch coupler, as described in U.S. Pat. No. 5,412,744 and illustrated in FIG. 3(c), which splits the input single mode field into two peaks. Another technique is the adiabatic mode shaper structure described in our pending UK patent application No. 0114494.8 which uses an extra tapered waveguide disposed adjacent the or each input waveguide to convert the single peak field of the input waveguide to a double peak field.
However, when any of the above-described features (namely the MMI, parabolic horn, multimode waveguide, Y-branch coupler or other passband flattening structure) are employed for the purpose of passband flattening, it is found that AWGs fabricated according to these designs in practice suffer from asymmetry in the passbands of different wavelength output channels. This is illustrated in FIG. 4 which shows that although the passbands P1,P2,P3,P4 (plotted as the Transmission, in dB, against wavelength, xcex) of the central four output channels of the AWG are generally symmetrical as expected, the passbands of the other channels become asymmetrical to the right and left of these central four, the asymmetry in the channels to the right being generally inverse to the asymmetry in the channels to the left. This asymmetry is believed to be caused by off-axis aberrations (sometimes referred to as xe2x80x9cCOMAxe2x80x9d) in the second slab waveguide 4. The further the optical signal condenses away from the aiming point C1 of the array waveguides on the output face 20 of the slab waveguide 4, the greater the asymmetry in the passband becomes. One undesirable effect of this asymmetry is that it causes undesirable fluctuation in insertion loss with variation in wavelength of the input optical signal.
This asymmetry effect in the flattened passband is also noted in Published Japanese Patent Application No. Hei 11-180118. The proposed solution in this patent application involves varying gradually the relative angular spacing, at the slab/array interface, of the array waveguides across the array so as to empirically correct for the off-axis aberrations which cause the asymmetry in the passband. However, the given formula does not provide an exact solution i.e. it does not completely remove the COMA aberration.
The existence of COMA in phasars, and the need to remove it is acknowledged and described in detail in xe2x80x9cIntegrated Optics: Design and Modellingxe2x80x9d by Reinhard Marz, Copyright 1995, by Artech House, Inc, ISBN 0-89006-668-X., Chapter 8, in particular pages 277-280. It is noted by Marz on page 279 that phased arrays based on Rowland mountings as described therein, have no COMA. According to Marz, the groove function of a Rowland mounting must be linear along the tangent to the grating line of the phasar. As illustrated in FIG. 5, this means that the lateral spacing xcex94y between adjacent array waveguides at the array slab interface (i.e. the grating line), measured along the tangent to the grating line (the Y-axis in FIG. 5), is uniform across the array, and that the angular spacing xcex94xcex8i of the adjacent array waveguides (on the grating line) varies gradually across the array, such that xcex94xcex8i greater than xcex94xcex80 where xcex94xcex80 is the angular spacing of the central two array waveguides. This is contrary to the teaching of all the well-known papers and patents relating to phasar design, such as the Smit review paper mentioned above) and the first patents relating to phasars e.g. U.S. Pat. No. 5,002,350 (Dragone), U.S. Pat. No. 5,243,672 (Dragone), which have always taught that the array waveguides should always be equally spaced, both angularly and laterally, on the grating line (which is the slab/array waveguides interface). Marz""s phasar designs are derived from blazed grating theory, which he has then applied to phasar design. He does not disclose the type of AWG design proposed by Smit and Dragone in which the lateral and angular spacing of the arrayed waveguides on the grating line is constant across the array, as described herein with reference to FIG. 1, nor address passband flattening in such AWGs and the problem of asymmetry in the flattened passband due to the presence of COMA, nor does he address the problem of how to remove COMA from AWGs of this type.
It is an aim of the present invention to avoid or minimize one or more of the foregoing disadvantages.
According to a first aspect of the present invention there is provided an arrayed waveguide grating (AWG) comprising:
first and second slab waveguides and an array of waveguides optically coupled between the first slab waveguide and the second slab waveguide, the array waveguides having predetermined optical path length differences therebetween, and each said slab waveguide having a first side which is optically coupled to the array waveguides and which has a first radius of curvature, and an opposing second side which has a second radius of curvature which is substantially half the magnitude of the first radius of curvature; and
a plurality of output waveguides optically coupled to the second side of the second slab waveguide; wherein
at the first side of the second slab waveguide the array waveguide angle xcex8i of the ith array waveguide, with respect to a straight line passing through the centres of curvature of the first and second sides of the second slab waveguide, is defined substantially by the following equation:
xcex8i=ArcSine (i.xcex94xcex8), where i=xe2x88x92(Nxe2x88x921)/2, xe2x88x92(Nxe2x88x921)/2+1, . . . , +(Nxe2x88x921)/2 
where i is the array waveguide number, N is the number of array waveguides, and xcex94xcex8 is a constant. This constant xcex94xcex8 may be substantially equal to the angular spacing, at the first side of the second slab waveguide, of the two array waveguides at the centre of the array.
In this case the third order aberration (COMA) from the second slab waveguide is substantially removed. Thus, there is no longer any asymmetry (or in practice no significant asymmetry) in the AWG channel outputs due to COMA from the second slab waveguide.
The array waveguide angle xcex8i may be defined exactly by the above equation:
xcex8i=ArcSine (i.xcex94xcex8), where i=xe2x88x92(Nxe2x88x921)/2, xe2x88x92(Nxe2x88x921)/2+1, . . . , +(Nxe2x88x921)/2. 
Alternatively, the function ArcSine (i.xcex94xcex8) in the above equation may be approximated by, for example, taking the third order (or a higher order) Taylor expansion of the function, whereby the array waveguide angle xcex8i is defined by the following equation: xcex8i=ixcex94xcex8+⅙ (i3.xcex94xcex83). With this approximation, the array waveguide angle xcex8i is still substantially defined by the function ArcSine (i.xcex94xcex8). Due to the presence of the third order term, the COMA will still be corrected.
Even if the function xcex8i(i) contains higher order terms, this will not affect removal of the COMA. Thus, according to another aspect of the invention there is provided an AWG comprising first and second slab waveguides and an array of waveguides optically coupled between the first slab waveguide and the second slab waveguide, the array waveguides having predetermined optical path length differences therebetween, and each said slab waveguide having a first side which is optically coupled to the array waveguides and which has a first radius of curvature, and an opposing second side which has a second radius of curvature which is substantially half the magnitude of the first radius of curvature; and a plurality of output waveguides optically coupled to the second side of the second slab waveguide; wherein at the first side of the second slab waveguide the array waveguide angle xcex8i of the ith array waveguide, with respect to a straight line passing through the centres of curvature of the first and second sides of the second slab waveguide, is defined substantially by a function xcex8i(i), where:
xcex8i(i)=ixcex94xcex8+a.i2+⅙(i3.xcex94xcex83)+b.i4+c.i5+d.i6+. . . , 
where i is the array waveguide number, xcex94xcex8 is a constant, and a,b,c,d, . . . are all constants. One or more of a,b,c,d . . . may be zero.
Preferably, the output waveguides are substantially single-mode waveguides and the AWG further includes at least one substantially single-mode input waveguide optically coupled to the second side of the first slab waveguide, and passband flattening means disposed between at least one of the substantially single mode input and output waveguides and an adjacent one of the first and second slab waveguides. Where there is only a single input waveguide, the optical axis of the input waveguide, and of any passband flattening means coupled thereto, is preferably aligned with the centres of curvature of the first and second sides of the first slab waveguide. In this case, there will be substantially no asymmetry in the channel output signals due to COMA from the first slab waveguide. Since the COMA from the second slab waveguide has already been removed, a uniform (or in practice substantially uniform) flattened passband shape across all the output channels of the AWG is achieved.
Alternatively, if the optical axis of the input waveguide, and any passband flattening means coupled thereto, does not intersect with the second side of the first slab at the centre of curvature of the first side of the first slab (we refer to this as an off-centre input waveguide), then there will be asymmetry in the demultiplexed signals in the AWG output channels which is due to COMA from the first slab waveguide, unless the angular spacing of the array waveguides at the first slab is also chirped to remove this COMA. Thus, in the AWG according to the above-described first aspect of the invention, the array waveguide angle xcex8i of the ith array waveguide, at the first side of the first slab waveguide, with respect to a straight line passing through the centres of curvature of the first and second sides of the first slab waveguide, is preferably defined substantially by the equation:
xcex8i=ArcSine (i.xcex94xcex8), where i=xe2x88x92(Nxe2x88x921)/2, xe2x88x92(Nxe2x88x921)/2+1, . . . , +(Nxe2x88x921)/2 
where i is the array waveguide number, N is the number of array waveguides, and xcex94xcex8 is a constant which may be substantially equal to the angular spacing, at the first side of the first slab waveguide, of the two array waveguides at the centre of the array. In this AWG there will be substantially no COMA present from either the first or second slab waveguide, even if an off-centre input waveguide is used. Where there is more than one input waveguide, any of the input waveguides can thus be used to input the multiplexed input signal to the first slab, without introducing asymmetry in the AWG output channel signals due to COMA from the first slab waveguide. Again, the function ArcSine (i.xcex94xcex8) may be approximated to its third order Taylor expansion, namely ixcex94xcex8+⅙(i3.xcex94xcex83).
The passband flattening means may comprise any suitable feature which causes flattening of the normally Gaussian-shaped passband of the AWG. For example, the passband flattening means may comprise a multi-mode interferometer (MMI), a parabolic horn, a Y-branch coupler, an adiabatic mode shaper as described in our pending British Patent Application No. 0114494.8, or a tapered waveguide such as described in our pending British patent application No. 0114608.3. Thus, the passband flattening means may comprise a non-adiabatic tapered waveguide which widens in width towards the respective slab waveguide, and at least an initial portion of the non-adiabatic tapered waveguide, which initial portion is connected to said at least one of the substantially single mode input and output waveguides, has a taper angle which increases towards said respective slab waveguide, and the non-adiabatic tapered waveguide merges substantially continuously with said at least one of the substantially single mode input and output waveguides. Nevertheless, other passband flattening features may alternatively be used.
According to another aspect of the invention, there is provided a method of substantially avoiding asymmetry in the passband of at least some of the output channels of an AWG comprising: first and second slab waveguides and an array of waveguides optically coupled between the first slab waveguide and the second slab waveguide, the array waveguides having predetermined optical path length differences therebetween, and each said slab waveguide having a first side which is optically coupled to the array waveguides and which has a first radius of curvature, and an opposing second side which has a second radius of curvature which is substantially half the magnitude of said first radius of curvature; and a plurality of output waveguides optically coupled to the second side of the second slab waveguide, for outputting different output channels of the AWG; wherein the method comprises chirping the angular spacing of the array waveguides at the first side of at least the second slab waveguide so that the array waveguide angle xcex8i of the ith array waveguide, with respect to a straight line passing through the centres of curvature of the first and second sides of the second slab waveguide, is defined substantially by the following equation:
xcex8i=ArcSine (i.xcex94xcex8), where i=xe2x88x92(Nxe2x88x921)/2, xe2x88x92(Nxe2x88x921)/2+1, . . . , +(Nxe2x88x921)/2 
where i is the array waveguide number, N is the number of array waveguides, and xcex94xcex8 is a constant which may be substantially equal to the angular spacing, at the first side of the second slab, of the two array waveguides at the centre of the array. The ArcSine function may, in practice, be approximated to its third order Taylor expansion.
Embodiments of the invention will now be described by way of example only arid with reference to the accompanying drawings in which: